For a horizontal asymptote we look at what happens to the function as x ---> infinity.

Use an intuitive approach ...
as x becomes very large, say x = 1 million, the denominator becomes large much faster than the numerator.
So you have a division by a hugely large number resulting in a number close to zero
So when x ---> + infinity, f(x) ---> +0 (still above the x-axis)
wen x ---> - infinity , f(x) ----> -0 (slightly below the x-axis)